Vol=
w dx dy and we see that by comparison with Eq. (3.8)
T =2 Vol
The analogy therefore provides an extremely useful method of analysing torsion bars possessing irregular cross-sections for which stress function forms are not known.
Hetényi2describes experimental techniques for this approach. In addition to the strictly experimental use of the analogy it is also helpful in the visual appreciation of a particular torsion problem. The contour lines often indicate a form for the stress function, enabling a solution to be obtained by the method of Section 3.1. Stress concentrations are made apparent by the closeness of contour lines where the slope of the membrane is large.
These are in evidence at sharp internal corners, cut-outs, discontinuities, etc.
3.4 Torsion of a narrow rectangular strip
In Chapter 18 we shall investigate the torsion of thin-walled open section beams; the development of the theory being based on the analysis of a narrow rectangular strip subjected to torque. We now conveniently apply the membrane analogy to the torsion of such a strip shown in Fig. 3.9. The corresponding membrane surface has the same cross-sectional shape at all points along its length except for small regions near its ends where it flattens out. If we ignore these regions and assume that the shape of the
Fig. 3.9Torsion of a narrow rectangular strip.
membrane is independent of y then Eq. (3.11) simplifies to d2φ
dx2 = −2Gdθ dz Integrating twice
φ= −Gdθ
dzx2+Bx+C
Substituting the boundary conditionsφ=0 at x= ±t/2 we have φ= −Gdθ
dz
x2− t
2 2
(3.26) Althoughφdoes not disappear along the short edges of the strip and therefore does not give an exact solution, the actual volume of the membrane differs only slightly from the assumed volume so that the corresponding torque and shear stresses are reasonably accurate. Also, the maximum shear stress occurs along the long sides of the strip where the contours are closely spaced, indicating, in any case, that conditions in the end region of the strip are relatively unimportant.
The stress distribution is obtained by substituting Eq. (3.26) in Eqs (3.2), then τzy =2Gxdθ
dz τzx=0 (3.27)
the shear stress varying linearly across the thickness and attaining a maximum τzy,max= ±Gtdθ
dz (3.28)
at the outside of the long edges as predicted. The torsion constant J follows from the substitution of Eq. (3.26) into (3.13), giving
J= st3
3 (3.29)
and
τzy,max= 3T st3
These equations represent exact solutions when the assumed shape of the deflected membrane is the actual shape. This condition arises only when the ratio s/t approaches infinity; however, for ratios in excess of 10 the error is of the order of only 6 per cent.
Obviously the approximate nature of the solution increases as s/t decreases. Therefore, in order to retain the usefulness of the analysis, a factoràis included in the torsion constant, i.e.
J = àst3 3
Values ofàfor different types of section are found experimentally and quoted in various references.3,4We observe that as s/t approaches infinityàapproaches unity.
References 81
Fig. 3.10Warping of a thin rectangular strip.
The cross-section of the narrow rectangular strip of Fig. 3.9 does not remain plane after loading but suffers warping displacements normal to its plane; this warping may be determined using either of Eqs (3.10). From the first of these equations
∂w
∂x =ydθ
dz (3.30)
sinceτzx=0 (see Eqs (3.27)). Integrating Eq. (3.30) we obtain w=xydθ
dz +constant (3.31)
Since the cross-section is doubly symmetrical w=0 at x=y=0 so that the constant in Eq. (3.31) is zero. Therefore
w=xydθ
dz (3.32)
and the warping distribution at any cross-section is as shown in Fig. 3.10.
We should not close this chapter without mentioning alternative methods of solution of the torsion problem. These in fact provide approximate solutions for the wide range of problems for which exact solutions are not known. Examples of this approach are the numerical finite difference method and the Rayleigh–Ritz method based on energy principles.5
References
1 Wang, C. T., Applied Elasticity, McGraw-Hill Book Company, New York, 1953.
2 Hetényi, M., Handbook of Experimental Stress Analysis, John Wiley and Sons, Inc., New York, 1950.
3 Roark, R. J., Formulas for Stress and Strain, 4th edition, McGraw-Hill Book Company, New York, 1965.
4 Handbook of Aeronautics, No. 1, Structural Principles and Data, 4th edition. Published under the authority of the Royal Aeronautical Society, The New Era Publishing Co. Ltd., London, 1952.
5 Timoshenko, S. and Goodier, J. N., Theory of Elasticity, 2nd edition, McGraw-Hill Book Company, New York, 1951.
Problems
P.3.1 Show that the stress functionφ=k(r2−a2) is applicable to the solution of a solid circular section bar of radius a. Determine the stress distribution in the bar in terms of the applied torque, the rate of twist and the warping of the cross-section.
Is it possible to use this stress function in the solution for a circular bar of hollow section?
Ans. τ=Tr/Ip where Ip=πa4/2,
dθ/dz=2T/Gπa4, w=0 everywhere.
P.3.2 Deduce a suitable warping function for the circular section bar of P.3.1 and hence derive the expressions for stress distribution and rate of twist.
Ans. ψ=0, τzx= −Ty
Ip, τzy=Tx
Ip, τzs=Tr Ip, dθ
dz = T GIP
P.3.3 Show that the warping functionψ=kxy, in which k is an unknown constant, may be used to solve the torsion problem for the elliptical section of Example 3.2.
P.3.4 Show that the stress function φ= −Gdθ
dz 1
2(x2+y2)− 1
2a(x3−3xy2)− 2 27a2
is the correct solution for a bar having a cross-section in the form of the equilateral triangle shown in Fig. P.3.4. Determine the shear stress distribution, the rate of twist and the warping of the cross-section. Find the position and magnitude of the maximum shear stress.
Fig. P.3.4
Problems 83
Ans. τzy=Gdθ
dz
x−3x2 2a +3y2
2a
τzx= −Gdθ dz
y+3xy
a
τmax(at centre of each side)= −a 2Gdθ
dz dθ
dz = 15√ 3T Ga4 w= 1
2a dθ
dz(y3−3x2y).
P.3.5 Determine the maximum shear stress and the rate of twist in terms of the applied torque T for the section comprising narrow rectangular strips shown in Fig.
P.3.5.
Fig. P.3.5
Ans. τmax=3T/(2a+b)t2, dθ/dz=3T/G(2a+b)t3.